Problem in finite differences

Show that the nth divided difference of is 1
solution:
We have,dash indicating that the term is to be excluded from denominator
I don't understand this step. Can any math helper,math expert explain me? Especially 1 on R. H. S.

Re: Problem in finite differences

Originally Posted by chiro

Hey Vinod.

What is the dummy variable involved for the summation? Is the same variable within the expression directly to the right of sigma?

Hey Chiro,
As per my understanding,L. H. S.-the term next to summation'(sigma') in R. H. S. is equal to 1 as stated in the problem.I think it is the nth divided difference of n arguments()
Next steps in the solution are as follows:Put x=0 on both sides, we have 0=1 - f{}

Re: Problem in finite differences

Hey chiro,

Your reply suggests me that you are incognizant of divided difference properties. The term next to sigma' is based on the property which states that divided difference is a symmetrical function of all the arguments involved.

Hence, the LHS and the RHS are equal when $n=1$. To prove it is true for all $n$, you can attempt to use induction. You know it is true when $n=0$ (and when $n=1$), so assume it is true when $n=k$ and try to prove it for $n=k+1$.